The Non-Commutative Geometry of Algebraic Integers

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In one of the main outcomes of the PIMS CRG20 Operator Algebras and Non-Commutative Geometry, Joachim Cuntz, Christopher Deninger and Marcelo Laca associated Toeplitz-type C*-algebras to the rings of integers of algebraic number fields. In earlier work Cuntz and Li had produced purely infinite simple C*-algebras from rings of algebraic integers, and Laca and Raeburn had introduced a Toeplitz type C*-algebra for the natural numbers. The C*-algebras of [CDL] are simultaneous generalizations of the latter and extensions of the former; they have the right functorial properties, and thus the potential for generating invariants for number fields. Moreover, the natural non-commutative (quantum) dynamical systems based on the C*-algebras from [CDL] have equilibrium states exhibiting phase transitions that point to a connection with class field theory. This work has already had a significant impact in the area, sparking further research in several directions. One of the most interesting new directions is the study of C*-algebraic invariants for number fields that combine K-theory and ideal structure, appearing in subsequent work of Cuntz-Li, Li, and Echterhoff-Laca. CRG20 fostered the generation of key ideas and attracted new researchers to the interface between operator algebras and number theory; the increased level of activity in this area prompted a focused workshop in Münster December 2011, an Oberwolfach workshop in April 2012, and a BIRS workshop taking place in 2013.